Yves Hellegouarch

Few problems have generated as much mathematics as FLT: Fermat's Last Theorem. And since the Theorem has been proven, mathematicians have written much about it. There are some good books regarding the history of the problem: The Fermat Diary, Fermat's Enigma; and some which deal with the basic mathematics of FLT: Fermat's Last Theorem for Amateurs. Yves Hellegouarch's book, Invitation to the Mathematics of Fermat-Wiles, blends the two, but mainly focuses on the more advanced mathematics of FLT. The original text was in French; the English translation is provided by Leila Schneps, who is no stranger to the mathematics of FLT. There are some minor problems with the translation (mostly having to do with quantifiers): part (i) of Theorem 1.6.4, the sentence after Theorem 4.6.1, the word premier appearing in the Euler product for the Riemann zeta function. As for the setup of the book, it has a short foreword, followed by six chapters, an appendix, and a bibliography with just over one hundred references. Each chapter has several sections, followed by a brief commentary by Hellegouarch, and then an extensive list of problems and exercises (while the text of the six chapters covers roughly 246 pages, the exercises cover 117 pages).

The first chapter gives some history of FLT and some of the early developments. Much discussion is devoted to Fermat's method of infinite descent. We are then led to Kummer's idea of factorization in the ring of cyclotomic integers. The exercises for chapter 1 involve the method of infinite descent, the study of nontrivial solutions of "Fermat-like" equations such as x4 - y4 = z2 and x3 + y3= 2z3, a proof of Mason's Theorem on the parametrizability of curves, the algebra of quadratic fields, and the geometry of lattices.

The second chapter introduces the idea of elliptic functions by way of elliptic integrals. The standard properties are derived, leading to lattices, the Eisenstein series, the Weierstrass cubic equation, and the group structure of the points on this cubic. The elliptic functions are then redone via the theory of loxodromic functions. The exercises for chapter 2 cover more properties of elliptic integrals, more results on the Eisenstein series, the Weierstrass p- function and sigma function, and some of Abel's work on elliptic functions.

In the third chapter, the author develops some of the algebra necessary for the proof of FLT: absolute values on Q, with particular emphasis on the field of p-adic numbers. Some basics of representation theory and Galois theory are given. The exercises develop the theory of absolute values (including a proof of Hensel's Lemma and Ostrowski's Theorem), the theory of group representations (with a proof of Schur's Lemma), and a proof of Hermite's Theorem on the transcendence of pi.

In the fourth chapter, we start focusing heavily on elliptic curves. Some of the basics of algebraic geometry are developed here: singular and non-singular points, birational equivalence, Bezout's Theorem, the Nine Point Theorem, the group structure of rational points on an elliptic curve, and the corresponding notion of n-division points on an elliptic curve. We also see some of the major results needed for Wiles' proof: Mazur's Theorem on the torsion subgroups of rational points on an elliptic curve and the Mordell-Weil Theorem. The study of the Weierstrass cubic equation leads to the modular invariant j (foreshadowing the topic of the next chapter) and the Hasse-Weil L-function of an elliptic curve. This chapter has the longest set of exercises; much of it has to do with algebraic geometry, and the last problem develops many of the standard results about congruent numbers (a la Koblitz).

Chapter 5 covers the topic of modular forms. Some history is given, with some familiar results involving partitions and the Jacobi theta functions. We then get the definition of modular forms for the modular group SL2(Z). These are defined by looking at the Eisenstein series and the modular function j for a lattice. Some of the basic facts are developed (modular forms over the Hecke subgroups of SL2(Z), the dimension of the space of modular forms of weight k, the structure of Mk (SL2(Z)), the Petersson inner product, the Hecke operators, and the L-series for a Hecke eigenform). We are given (without details) Wiles's 1994 proof of the Taniyama-Weil conjecture on the modular form associated to a semi-stable elliptic curve defined over Q. The exercise set covers Ramanujan's partition congruences, some results on the order of the congruence subgroups of SL2(Z), some properties of the Jacobi theta functions, and a proof of Hardy's theorem on the zeroes of the Riemann zeta function.

The final chapter deals with Wiles' attempt to put all of the ideas together. We get the "Frey-Hellegourach" elliptic curve EA,B,C associated to the equation A + B + C = 0. The properties of this curve allow us to examine an equation like xp + yp = zp for rational solutions. FLT is proven via Wiles' Theorem, which uses the Mazur-Ribet Theorem concerning Serre's conjectures. Very little is proven here, but the heuristic is fairly well described. The author also discusses some related Diophantine equations and the ABC conjecture. The exercises deal with some of these Diophantine equations and some consequences of the ABC conjecture.

Finally, there is a short appendix which gives a very general overview to the elliptic approach to solving FLT and some of the motivations behind the proof. As with many of the more advanced theorems and results in this book, there are no proofs. However, the relevant references are given. As an introduction to the mathematics necessary for Wiles' proof (algebra, p-adic analysis, algebraic geometry, elliptic curves and modular functions), this book succeeds. As a textbook, it might succeed for a graduate course specifically designed to introduce the mathematics of FLT. But it probably wouldn't work so well for any other kind of course. If you want to learn about algebra, try Hungerford; if you want p-adic analysis, try Koblitz or Gouvêa; if you're interested in algebraic geometry, go to Kunz or Hartshorne; if you need elliptic curves and/or modular forms, try Silverman or another Koblitz. This book doesn't go very deeply into these areas--just enough to get what is needed for the proof of FLT. But it does that reasonably well. And, along the way, it provides some interesting history, including some correspondences and works from Jakob Bernoulli, Neils Abel, and Colin Maclaurin.